We consider a class of two-dimensional Schrödinger operator with a singular interaction of the delta type and a fixed strength beta supported by an infinite family of concentric, equidistantly spaced circles, and discuss what happens below the essential spectrum when the system is amended by an Aharonov–Bohm flux alpha in [0,frac{1}{2}] in the center. It is shown that if beta ne 0, there is a critical value alpha _{mathrm {crit}}in (0,frac{1}{2}) such that the discrete spectrum has an accumulation point when alpha <alpha _{mathrm {crit}}, while for alpha ge alpha _{mathrm {crit}} the number of eigenvalues is at most finite, in particular, the discrete spectrum is empty for any fixed alpha in (0,frac{1}{2}) and |beta | small enough.